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Some word problems lead us to inequalities rather than equations, which is good, because algebra would be awfully boring if we never got to mix it up. We know what you're thinking, but don't say it.

When we're solving an inequality to answer a word problem, we need to think about the answer before we write it down, especially for problems where answers need to be integers.

Sample Problem

Liam has $20. He wants to buy some comic books that cost $1.75 each. He doesn't like comic books, but his mom is forcing him to start a hobby so he doesn't spend all day, every day watching television. How many comic books can he afford to buy?

Liam only has $20, so we need this inequality to be true:

(1.75)(the number of comic books Liam buys) ≤ 20

The above is true because we're assuming Liam won't be able to sweet-talk his way into any additional comic books, and that he won't go back into his wallet for a credit card. Probably a safe assumption to make.

Let x be the number of comic books Liam buys, and translate the inequality into symbols as

(1.75)x ≤ 20

Solving this inequality gives us some horrible non-integer between 11 and 12. It is approximately 11.4, but actually a number that causes much more of a headache to look at. Liam can't buy 12 comic books, because he doesn't have enough money. He also can't buy part of a comic book, as they usually won't let you buy individual pages, so he can buy 11 comic books. Plus, he'll still have a few cents leftover to throw into a fountain and wish he were back at home watching television.

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